We address the problem of sparsity-promoting optimal control of cyber-physical systems (CPSs) in the presence of communication delays. The delays are categorized into two types - namely, an inter-layer delay for passing state and control information between the physical layer and the cyber layer, and an intra-layer delay that operates between the computing agents, referred to here as control nodes (CNs), within the cyber-layer. Our objective is to minimize the closed-loop H2-norm of the physical system by co-designing an optimal combination of these two delays and a sparse state-feedback controller while respecting a given bandwidth cost constraint. We propose a two-loop optimization algorithm for this. Based on the alternating directions method of multipliers (ADMM), the inner loop handles the conflicting directions between the decreasing H2-norm and the increasing sparsity level of the controller. The outer loop comprises a semidefinite program (SDP)-based relaxation of non-convex inequalities necessary for closed-loop stability. Moreover, for CPSs where the state and control information assigned to the CNs are not private, we derive an additional algorithm that further sparsifies the communication topology by modifying the row and column structures of the obtained controller, resulting in reassigning the communication map between the cyber and physical layers, and determining which physical agent should send its state information to which CN. Proofs for closed-loop stability and optimality are provided for both algorithms, followed by numerical simulations.